I need to find the volume of a frustum of a pyramid with square base of side $b$, square top of side $a$, and height $h$(using integrals). I have no idea how to do questions like these, I only know to use the disc/washer/cylindrical shells methods and rotate a region around any line and find its volume. For these type of question, I find myself at a loss as to where to even start. Any hints on general about starting these kind of problems are also appreciated!
[Math] Finding volume of a frustum of a pyramid
calculusintegrationvolume
Best Answer
A volume element is
$$dV = A(y) dy$$
where the sides in the square cross-sectional area $A(y)$ behaves linearly with height:
$$A(y) = \left[ b-\frac{y}{h} (b-a)\right ]^2 = b^2-2 b (b-a)\frac{y}{h} +\frac{(b-a)^2} {h^2} y^2$$
so the integral is
$$V = \int_0^h dy \, A(y) = b^2 h - b (b-a) h +\frac13 (b-a)^2 h$$
or, simplifying,
$$V = \frac13\frac{b^3-a^3}{b-a} h $$